# Law of Cosines Calculator

## This Calculation Equation & Triangle

\[ A = \cos^{-1} \left[ \frac{b^2+c^2-a^2}{2bc} \right] \]

A = angle A

B = angle B

C = angle C

a = side a

b = side b

c = side c

P = perimeter

s = semi-perimeter

K = area

r = radius of inscribed circle

R = radius of circumscribed circle

*Length units are for your reference-only since the value of the resulting lengths will always be the same no matter what the units are.

## Calculator Use

Uses the law of cosines to calculate unknown angles or sides of a triangle. In order to calculate the unknown values you must enter 3 known values.

To calculate any angle, A, B or C, enter 3 side lengths a, b and c. This is the same calculation as Side-Side-Side (SSS) Theorem. To calculate side a for example, enter the opposite angle A and the two other adjacent sides b and c. Using different forms of the law of cosines we can calculate all of the other unknown angles or sides. This is the same calculation as Side-Angle-Side (SAS) Theorem.

## Law of Cosines

If a, b and c are the lengths of the legs of a triangle opposite to the angles A, B and C respectively; then the law of cosines states:

\[ a^2 = b^2 + c^2 - 2bc \cos A \] \[ b^2 = a^2 + c^2 - 2ac \cos B \] \[ c^2 = a^2 + b^2 - 2ab \cos C \]### Law of Cosines solving for sides a, b, and c

\[ a = \sqrt{b^2 + c^2 - 2bc \cos A } \] \[ b = \sqrt{a^2 + c^2 - 2ac \cos B } \] \[ c = \sqrt{a^2 + b^2 - 2ab \cos C } \]### Law of Cosines solving for angles A, B, and C

\[ A = \cos^{-1} \left[ \frac{b^2+c^2-a^2}{2bc} \right] \] \[ B = \cos^{-1} \left[ \frac{a^2+c^2-b^2}{2ac} \right] \] \[ C = \cos^{-1} \left[ \frac{a^2+b^2-c^2}{2ab} \right] \]## Triangle Characteristics

Triangle perimeter, P = a + b + c

Triangle semi-perimeter, s = 0.5 * (a + b + c)

Triangle area, K = √[ s*(s-a)*(s-b)*(s-c)]

Radius of inscribed circle in the triangle, r = √[ (s-a)*(s-b)*(s-c) / s ]

Radius of circumscribed circle around triangle, R = (abc) / (4K)

## References/ Further Reading

Weisstein, Eric W. "Law of Cosines" From *MathWorld*-- A Wolfram Web Resource. Law of Cosines.

Zwillinger, Daniel (Editor-in-Chief). *CRC Standard Mathematical Tables and Formulae, 31st Edition* New York, NY: CRC Press, p. 512, 2003.

http://hyperphysics.phy-astr.gsu.edu/hbase/lcos.html

http://hyperphysics.phy-astr.gsu.edu/hbase/lsin.html

**Cite this content, page or calculator as:**

Furey, Edward "Law of Cosines Calculator"; from *http://www.calculatorsoup.com* - Online Calculator Resource.